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One-Step and Two-Step Estimation of the Effects of Exogenous Variables on Technical Efficiency Levels
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## TL;DR
This paper compares one-step and two-step estimation for stochastic frontier models when exogenous variables affect technical inefficiency. In Monte Carlo simulations, the one-step estimator performs well, while the two-step procedure shows serious bias in both the first and second stages.
## Key findings
- In the base case simulation (200 replications, sample size \(N=200\)), the one-step estimator is close to unbiased: \(\hat\beta=-0.0010\), \(\hat\delta=1.0010\), and average estimated efficiency \(E(\hat e^{-u})=0.5141\) versus the true \(0.5165\).
- Under the same base case, the two-step estimator is badly biased: \(\hat\beta=-0.3064\), estimated efficiency \(E(\hat e^{-u})=0.4482\) versus true \(0.52\) (table reports \(0.5165\) in the design and about \(0.52\) in results discussion), and the second-step estimate of \(\delta\) is \(0.3539\) versus the true value \(1\).
- The paper reports that the two-step bias persists even when \(x\) and \(z\) are independent: for \(\rho=0\), the second-step estimate of \(\delta\) is still \(0.4069\), and the estimated efficiencies remain underdispersed and downward biased.
- Increasing sample size from \(N=200\) to \(N=500\) and \(N=1000\) lowers standard deviations but does not remove the two-step biases; for example, the two-step \(\hat\delta\) stays around \(0.35\)–\(0.36\) while the true value is \(1\).
## Methods
- The paper studies a stochastic frontier model in which inefficiency depends on exogenous variables \(z\), emphasizing a “scaling property” specification where the one-sided inefficiency term is a function of \(z\) times a basic one-sided error.
- It uses Monte Carlo simulations with 2,000 replications per design point. The base case sets \(\beta=0\), \(\delta=1\), \(\rho=0.5\), \(\gamma=1\), \(\sigma_v=1\), \(\mu=0\), and \(N=200\).
- One-step estimation is maximum likelihood under the correctly specified model; two-step estimation first sets \(\delta=0\) in the frontier MLE, then regresses \(\ln(\hat u)\) on \(z\) to estimate \(\delta\).
## Limitations
- The paper’s own simulation evidence is limited to one simple model with the scaling property and a specific truncated-normal basic distribution; results may not generalize beyond that setup.
- The text makes evident, though it does not frame it as a formal limitation, that some parameters (\(\mu\) and \(\gamma\)) are “hard” to estimate and produced outliers severe enough that the authors truncated the most extreme 0.3% of replications for those parameters.
- Not stated in the paper.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Our simulations strongly confirm the relevance of this observation, since the two-step estimates of the effect of z on efficiency levels are seriously biased downward in all cases.”
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In the paper
“We first note that the one-step estimates look pretty good. In particular, there is no evidence of significant bias.”
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In the paper
“The parameters of the truncated normal base distribution ( µ and γ ) have rather large variances (their standard deviations are definitely not small relative to the parameter values themselves)”
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In the paper
“For the one-step estimator, the true value of δ is not terribly important.”
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In the paper
“But this does not seem to cause any problems for the parameters of main interest ( β , δ and the individual ri).”
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In the paper
“Second, the estimates of the ri (the technical efficiencies) are biased downward (mean = 0.45, compared to the mean of the actual ri of 0.52).”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Changing β has no substantive effects on our results.”
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In the paper
“First, the estimate of β is biased downward (mean = -0.31, compared to the true value = 0).”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Third, the second-step estimator of δ is very significantly biased downward (mean = 0.35, compared to the true value = 1).”
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In the paper
“Changing δ makes very little difference to the properties of the estimates of β , δ or σ v.”
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In the paper
“The bias of the estimate of β and of the two-step estimate of δ increases, and the estimates of the ri are also more biased, and less correlated with the true ri, compared to Case 1.”
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In the paper
“As expected, the bias of the first-step estimate of β is effectively zero when δ = 0, and it grows with δ .”
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In the paper
“All of these biases are in the expected direction, and the size of the biases is definitely not small.”
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In the paper
“They are less strongly correlated with the true ri and they are underdispersed, compared to the estimates from the one-step model.”
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In the paper
“The individual ri are seriously biased for δ = 1 but not for the smaller values of δ .”
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In the paper
“The two-step estimator of δ is biased for both δ = 1 and δ = 0.5, but the bias disappears as δ → 0.”
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“An interesting curiosity is the bias in the two-step estimator of δ based on the estimates of ui from the one-step model (mean estimate = 0.92, compared to the true value of δ = 1). This is not just finite-sample bias (it persists with larger sample sizes).”
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In the paper
“The mean of the estimate (both one-step and two-step) of β changes by the same amount that β is changed, so that the bias, standard deviation and MSE of the estimates of β are unchanged.”
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In the paper
“For this set of parameters, average technical efficiency is E(e -u ) = 0.5165.”
Strength:plain· not yet calibratedn = 200Tests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
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In the paper
“As δ → 0, our ability to estimate µ (and γ , to a lesser extent) deteriorates seriously.”
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In the paper
“The estimates of the individual ri become slightly more biased, and noticeably less strongly correlated with the true ri, when δ is small.”
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In the paper
“We find serious bias at all stages of this procedure. The size of the bias is very substantial and should argue strongly against two-step procedures.”
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“For the two-step estimates, surprisingly, imposing µ = 0 makes things worse (even though µ = 0 is a correct restriction).”
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In the paper
“As γ increases, the bias of the estimate of β increases, but the bias of the second-step estimate of δ decreases quite noticeably.”
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“Finally, the second-step estimate of δ is seriously biased even when ρ = 0, though its bias does grow with ρ .”
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In the paper
“The standard deviations of the estimators fall as N increases, but nothing else changes much.”
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In the paper
“For example, the mean of the two-step estimator of δ is 0.35 for σ v = 1, 0.12 for σ v = 3, and 0.05 for σ v = 5.”
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In the paper
“| 9 | µ = 1 | ˆ β | -0.0013 | 0.0963 | 0.0093 | | -0.3914 | 0.1191 | 0.1674 | |”
- Also stated in: § 4. CONCLUDING REMARKS · plain
“| 17 | N=500 | ˆ β | -0.0005 | 0.0572 | 0.0033 | | -0.3062 | 0.0661 | 0.0981 | |”
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- Also stated in: § 4. CONCLUDING REMARKS · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“| 9 | | E ( ˆ e - u ) | 0.3850 | 0.0294 | 0.0285 | 0.8387 | 0.3499 | 0.0197 | 0.0513 | 0.6891 |”
- Also stated in: § 4. CONCLUDING REMARKS · plain
“| 17 | N=500 | E ( ˆ e - u ) | 0.5160 | 0.0220 | 0.0338 | 0.8145 | 0.4470 | 0.0146 | 0.0615 | 0.6591 |”
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- Also stated in: § 4. CONCLUDING REMARKS · plain
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In the paper
“For the two-step estimators, we see more differences as µ changes, but they are still not really important or striking.”
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In the paper
“The true value of δ is one, so these are large biases indeed.”
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In the paper
“| 9 | E ( e - u ) | ˆ µ | 1.0591 | 0.3951 | 0.1595 | | -37.2499 | 1.1759 | 1464.4405 | |”
- Also stated in: § 4. CONCLUDING REMARKS · plain
“| 13 | E ( e - u ) | ˆ µ | 0.0429 | 0.4471 | 0.2016 | -37.2787 | 0.1688 | 1389.7299 | |”
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- Also stated in: § 4. CONCLUDING REMARKS · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“| 9 | | ˆ δ | 1.0002 | 0.0850 | 0.0072 | | - | - | - | |”
- Also stated in: § 4. CONCLUDING REMARKS · plain
“| 17 | N=500 | ˆ δ | 0.9995 | 0.0724 | 0.0052 | | - | - | - | |”
Strength:plain· not yet calibratedn = 500Tests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
- Also stated in: § 4. CONCLUDING REMARKS · plain
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In the paper
“The results are quite consistent with these expectations.”
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In the paper
“| 9 | | ˆ δ -2S | 0.9317 | 0.0888 | 0.0125 | 0.8025 | 0.4986 | 0.0566 | 0.2546 | 0.3469 |”
- Also stated in: § 4. CONCLUDING REMARKS · plain
“| 17 | N=500 | ˆ δ -2S | 0.9134 | 0.0736 | 0.0129 | 0.7911 | 0.3520 | 0.0336 | 0.4210 | 0.2375 |”
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- Also stated in: § 4. CONCLUDING REMARKS · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“| 17 | N=500 | ˆ γ | 1.0208 | 0.2393 | 0.0577 | | 53.4005 | 4.3493 | 2764.7151 | |”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one. The quote only partly states this claim.
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In the paper
“For the one-step estimators, the value of ρ is not important. The results are essentially identical across all these cases.”
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In the paper
“For larger γ , we estimate β a little worse, but we estimate δ and the individual ri a little better.”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
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“As µ moves from plus one to negative one, the estimates of β improve a little, and the estimates of the individual ri perhaps improve a little.”
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“As is often the case in simulations that involve numerical maximization, there were some problems with outliers, especially in estimation of γ and µ (the parameters of the truncated normal distribution).”
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In the paper
“This bias in the second-step regression, due to underdispersion in the estimates of u that do not take into account the effect of z on u, does not seem to be systematically discussed in the literature.”
What the sources say
- Not checked — no full text of the sourcecites Kumbhakar and Lovell (2000, p. 119) source
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In the paper
“That is, if the estimates of u are constructed ignoring the effect of z on σ u 2 , they will show less dependence on z than they should, and we should expect the second-step regressions to give downward biased estimates of the effect of z on u.”
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“Some but not all of the models in the literature have this property.”
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In the paper
“The one-step MLE will be based on the correctly specified model, and will therefore be consistent and asymptotically efficient.”
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In the paper
“For example, Caudill and Ford (1993) provide evidence on the bias of the estimated technological parameters, but not on the efficiency levels themselves or their relationship to the explanatory variables z.”
What the sources say
- Not checked — no full text of the sourcecites Caudill and Ford (1993) source
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In the paper
“For example, if h(z, δ ) = exp(z' δ ) and δ > 0, then large z will on average be associated with large u and also with large σ u 2”
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In the paper
“Presumably this is so because when δ is large, z correlates with u and is useful to help estimate u.”
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In the paper
“Saying the same thing, if we ignore the dependence of σ u 2 on z, we will shrink the observations with large u too much, and the observations with small u too little, and our estimates of u will be underdispersed.”
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“It is widely appreciated that the severity of the bias in estimation of the technological parameters (coefficients of x) depends on the magnitude of the correlation between x and z.”
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In the paper
“A second and less widely recognized problem is that the first-step technical efficiency measures are likely to be seriously underdispersed, so that the results of the second-step regression are likely to be biased downward. This is true regardless of whether x and z are correlated.”
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In the paper
“The first of these is that it captures the idea, which we find intuitively reasonable and appealing, that the shape of the distribution of u is the same for all firms.”
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In the paper
“That is, the second-step estimator of δ may be able to tell us whether or not z affects u (whether or not δ = 0) 3 but it cannot accurately estimate the effect of z on u when this effect exists.”
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“Models based on some other simple distributions, such as exponential, would also have the scaling property.”
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In the paper
“Our summary statistics are averages and are very sensitive to extreme outliers.”
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In the paper
“Importantly, this is true whether or not x and z are correlated.”
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In the paper
“One important implication of this viewpoint, which is sometimes missed, is that the variables z that determine inefficiency must not be functions of y.”
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In the paper
“This truncation of the results made very little difference for the parameters other than µ and γ .”
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In the paper
“Increasing σ v increases the amount of statistical noise in the model, and should increase the bias and/ or variance for each of the estimated parameters.”
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In the paper
“However, as long as δ is non-zero, this argument may be less than compelling, because the distribution of u depends on z whereas the distribution of v does not, and this is another way to distinguish u from v.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- Rests on an assumption: The distribution of u depends on z whereas the distribution of v does not.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Increasing γ while holding σ v constant has the effect of increasing the size of inefficiency relative to noise, and should tend to make it easier to estimate technical inefficiency precisely.”
Strength:hedged· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For example, when µ is large and positive, the degree of truncation is very small, and the shape of the distribution (if it can be separated from that of v) contains a lot of information about the parameters µ and σ 2 .”
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In this case the first-step estimator of β is unbiased, and we then estimate the ri under the correct assumption that they do not depend on the zi, so we ought to do well in this case.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As µ → ∞ , the distribution of u becomes normal and presumably becomes confounded with the normal distribution of v (statistical noise).”
Strength:hedged· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“When µ is large (in absolute value) but negative, on the other hand, we observe only the extreme right tail of the distribution, whose shape may not be very informative about µ and σ 2 , and the estimation problem may be harder just on that basis.”
Strength:hedged· not yet calibratedCausal:conditional· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“All in all, the two-step estimator performs well when the second step is not needed ( δ = 0) but is otherwise unreliable.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“These biases are substantial enough that we would recommend against using two-step procedures in any circumstances that we can envision.”
Load-bearingStrength:hedged· not yet calibratedTests✓ Asserted: passes✗ Checkable: fails✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim. It may be too vague to check against evidence.
The authors state this as an assumption.
In the theoretical exploration of underdispersion, x and z are independent.
“To explore this point more precisely, suppose that x and z are independent, so that the first-step regression is unbiased.”
The paper relies on this without support.
The distribution of u depends on z whereas the distribution of v does not.
“because the distribution of u depends on z whereas the distribution of v does not”
The paper relies on this without support.
The biases identified above are expected to persist asymptotically.
“the biases we have identified above are expected to persist asymptotically”
The paper relies on this without support.
The one-step estimators are based on a correctly specified model and are asymptotically optimal.
“The one-step estimators are based on a correctly specified model and are asymptotically optimal.”
Citations
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Saldías, R. (2010) · eDiss (Georg-August-Universität Göttingen) · doi:10.53846/goediss-1793
A study on the Technical Efficiency Effects of e-sale
노재확 (2011) · The e-Business Studies · doi:10.15719/geba.12.1.201103.311
Orea, L., & Steinbuks, J. (2011)
Specification of Merger Gains in the Norwegian Electricity Distribution IndustryPreprint
Saastamoinen, A., Bjørndal, E., & Bjørndal, M. (2016) · SSRN Electronic Journal · doi:10.2139/ssrn.2766936
Janoudi, S. (2014) · SSRN Electronic Journal · doi:10.2139/ssrn.2513249
Niaounakis, T. K., & Blank, J. L. T. (2015) · Data Archiving and Networked Services (DANS)
The Labor Allocation of Household Members and Farm Technical Efficiency in Chinese Agriculture
Yue, B., & 岳, 博. (2014) · Institutional Repositories DataBase (IRDB)
DOES URBAN PROXIMITY ENHANCE TECHNICAL EFFICIENCY? EVIDENCE FROM CHINESE AGRICULTUREPreprint
Mitterrand, F. (2013) · SSRN Electronic Journal
Alam, A. (2012) · Medical Entomology and Zoology
(Modeling of State Influence on the Processes of Economic Growth)Preprint
Shagas, N., Bozhechkova, A. V., Pervishin, Y. N., Perevyshina, E., & Tumanova, E. A. (2016) · SSRN Electronic Journal · doi:10.2139/ssrn.2752926
Shaik, S., & Mishra, A. K. (2010) · Advances in econometrics · doi:10.1108/s0731-9053(2010)0000026013
Production Efficiency Analysis of the Hungarian Meat Processing Industry
Berezvai, Z. (2015) · Competitio · doi:10.21845/comp/2015/1/2
Efficiency Analysis of the Chinese Banking Sector
Luo, D. (2016) · Palgrave Macmillan UK eBooks · doi:10.1057/9781137454669_2
An unbiased estimation of Wage frontiers
Álvarez, A. M. R. (2016) · Dialnet (Universidad de la Rioja)
Efficiency evaluation of Spanish hotel chainsPreprint
Wiper, M. P., Veiga, H., & Deng, Y. (2016) · RePEc: Research Papers in Economics
Capacitação Tecnológica e Desempenho Produtivo das Empresas Brasileiras
Silva, F. Q., & de Avellar, A. P. M. (2017) · Revista de Economia · doi:10.5380/re.v41i1.35794
Rice-Based Cropping Systems in the Delta of the Vu Gia Thu Bon River Basin in Central Vietnam
Pedroso, R., Tran, D. H., Trinh, V. Q., An, L. T., & Le, K. P. (2017) · Water resources development and management · doi:10.1007/978-981-10-2624-9_6
Owusu, R. (2017) · UWA Profiles and Research Repository (University of Western Australia) · doi:10.4225/23/59af416231e6f
“Later, Wang (2002) asserted that a combination of the two approaches is appropriate as it allows for testing non-monotonic relationship18 in the inefficiency and Z variables, as opposed to the application of either of the two approaches.”
BackgroundMethodResultInfluentialEssays on the study of production efficiency
Han, Y. (2008)
Koirala, K. H. (2015) · Louisiana State University · doi:10.31390/gradschool_dissertations.2208
“We applied a one-stage stochastic production frontier function as discussed by Wang and Schmidt (2002).”
BackgroundMethodInfluentialBanking Reform and Bank Performance in China
Jiang, C., & Yao, S. (2017) · Springer eBooks · doi:10.1007/978-3-319-63925-3_3
Khan, S. H., & Louis, L. (2017) · International Journal of Scientific Research in Mathematical and Statistical Sciences · doi:10.26438/ijsrmss/v4i6.16
“[7,2] assumed truncated normal distribution for the inefficiency term.[9] modelled the dependence of ) , ( z u on z by writing it as * ) , ( ) , ( u z h z u where u follows half normal distribution.”
The Effects of Multi-Dimensional Competition on Education Market Outcomes
Karakaplan, M. U. (2012) · OakTrust (Texas A&M University Libraries)
Junqueira, M. D. O. (2018) · Digital Library of Theses and Dissertations (Universidade de São Paulo) · doi:10.11606/t.8.2018.tde-15022018-120151
Willingness to Pay for Malaria Prophylaxis in Ethiopia
Soname, S., & Holloway, G. J. (2018) · Cooperative management · doi:10.1007/978-3-319-77122-9_7
Essays on Transaction Costs and Food Diversity in Developing Countries
David, C. S. (2018) · eDiss (Georg-August-Universität Göttingen) · doi:10.53846/goediss-6944
Essays on biased innovation and environmental policy
Stadler, B. (2018) · UEA Digital Repository (University of East Anglia)
Khataza, R. (2017) · UWA Profiles and Research Repository (University of Western Australia) · doi:10.4225/23/5a6181d9dfb6e
Belotti, F., & Ferrara, G. (2019) · SSRN Electronic Journal · doi:10.2139/ssrn.3414192
“(1996) and Ferrara and Vidoli (2017), produces biased results if the inefficiency determinants and the inputs are correlated (Wang and Schmidt, 2002).”
Bush, T. K., Hâu, T. V., Hiếu, T. S., et al. (2019) · Can Tho University Journal of Science · doi:10.22144/ctu.jvn.2019.076
Yane, S., & Berg, S. V. (2011) · SSRN Electronic Journal · doi:10.2139/ssrn.1907752
Daidone, S., & D’Amico, F. (2009) · SSRN Electronic Journal · doi:10.2139/ssrn.1434489
Haschka, R. E., Herwartz, H., & Schley, K. (2020) · doi:10.35841/public-health-policy.4.4.42-57
“Quantifying the production function and deviations from the efficient technology simultaneously, one avoids eventually invalid conclusions based on two-step approaches which process estimated first-step efficiency scores [31].”
MethodPuerta, A., & 'irez-Hassan, A. '. R. (2020) · RePEc: Research Papers in Economics
Berisso, O. (2020) · Journal of Development and Agricultural Economics · doi:10.5897/jdae2020.1167
“Lastly, following Wang and Schmidt (2002) the components of the TECs - Change due to passage of time (TECT), changes due to farm-characteristics (TECZ), changes due to environmental factors (TECW) and changes due to inputs (TECX) are computed respectively as:”
BackgroundMethodMilk production analysis with a flexible frontier production model
Niemi, T. (2020) · Työväentutkimus Vuosikirja
Chu, P., Xie, G., & Liu, Z. (2020) · International Journal of Economics and Finance · doi:10.5539/ijef.v12n10p105
“Wang and Schmidt (2002) using Monte Carlo method also confirmed that the one-step estimation method is better than the two-step method.”
MethodWassongma, H., & Fall, F. S. (2020) · Revue d économie du développement · doi:10.3917/edd.334.0097
Informal institutions, transaction risk, and firm productivity in Myanmar
Danquah, M., & Kunal Sen, A. (2020) · Working Paper Series · doi:10.35188/unu-wider/2020/811-5
“The heteroscedastic stochastic frontier models are important to this exercise because ignoring the heterogeneity in the stochastic frontier production model may cause biased estimates of the frontier function parameters as well as estimates of inefficiency (see Greene 2004; Hadri et al. 2003; Kumbhakar and Lovell 2000…”
BackgroundRisk Management and Efficiency of Microfinance Institutions
Zamore, S. (2018) · NORA - Norwegian Open Research Archives
Back Matter: Appendices A and B
Zhang, F. (2018) · World Bank, Washington, DC eBooks · doi:10.1596/978-1-4648-1154-8_bm
Three Essays on “Production and Technical Efficiency”
Mugabe, D. (2020) · doi:10.33915/etd.7748
School expenditure leakage and efficiency : the case of Thai compulsory education
Thasayaphan, J. (2010) · NRCT Data Center · doi:10.14457/nida.the.2010.47
Review of Modern Approaches for Assessing the Effectiveness of Banking
Леонов, М. В. (2021) · Journal of Applied Economic Research · doi:10.15826/vestnik.2021.20.2.013
Efficiency Effects in a Copula Based Stochastic Frontier Model
Yamaka, W. (2021) · Studies in computational intelligence · doi:10.1007/978-3-030-77094-5_11
Individual differences matter : personality traits and preferences in smallholders’ farm management
Qian, C. (2021) · doi:10.18174/550618 · Cited by 2
Alvarado, E., Brümmer, B., & Ibáñez, M. (2021) · AGROFOR · doi:10.7251/agreng2101096i
Productivity Spillovers From Foreign Direct Investment To Ethiopian Manufacturing SectorPreprint
Wajebo, T. W. (2021) · Research Square · doi:10.21203/rs.3.rs-903290/v1
“Econometric Model Specification Many studies used an augmented Cobb Douglas production function to overcome the misspecification problem because of it provides efficient estimates (Wang and Schmidt, 2002; De Mello, 1999).”
MethodZhao, S., & Parmeter, C. F. (2022) · SSRN Electronic Journal · doi:10.2139/ssrn.4125931
Nguyễn, T. T., Võ, H. T., & Lê, T. S. (2022) · Can Tho University Journal of Science · doi:10.22144/ctu.jvn.2022.213
Ye, Y., & Tao, Q. (2022) · Research Square · doi:10.21203/rs.3.rs-2207794/v1
Aktivitas Ekspor dan Inefisiensi Teknis Industri Andalan Ekspor Indonesia
Program Pascasarjana Ilmu Ekonomi, F. E. D. B. U. I., Rustiawati, R. B., Lubis, A. F., et al. (2019) · Jurnal Ekonomi dan Pembangunan Indonesia · doi:10.21002/jepi.2019.14
Productivity in Procurement Auctions of Pavement Contracts in Mexico
Prudencio, D. (2023) · Journal of Productivity Analysis · doi:10.1007/s11123-023-00677-0
Kumbhakar, S. C., Sickles, R. C., & Wang, H. (2023) · Empirical Economics · doi:10.1007/s00181-023-02438-2
“Among the many additional contributions Peter has made in this field of study, Lovell points out two additional seminalworks that focus on “measurable and policy-sensitive determinants of efficiency,” Wang and Schmidt (2002) and Alvarez et al. (2006).”
BackgroundSelf-citationLinguistic clustering and aggregate productive efficiency in Indonesia
Repkine, A. (2023) · Applied Economic Analysis · doi:10.1108/aea-04-2022-0124
Marcomini, G. R. (2023) · Informe GEPEC · doi:10.48075/igepec.v27i2.30543 · Cited by 1
Ouyang, X., Li, C., & Li, D. (2023) · Frontiers in artificial intelligence and applications · doi:10.3233/faia230176
黃台心, 黃., Huang, T., Lin, 曾. Y.-C., & Tseng, 鄭. C.-J. (2023) · 應用經濟論叢 · doi:10.53106/054696002023060113002
Blank, J. L. T., van Heezik, A. A. S., & Blank, B. (2023) · Public Sector Economics · doi:10.3326/pse.47.3.2
Eficiência na produção de batata
Marcomini, G. R. (2021) · doi:10.47749/t/unicamp.2021.1161707
Does Education Help Local Economies Reach Economic Potential? Evidence from South Africa
Mazorodze, B. T., & Maduku, H. (2023) · Athens Journal of Business & Economics · doi:10.30958/ajbe.10-1-4
“This approach is biased as the model estimated in the first stage is misspecified (Wang and Schmidt 2002).”
MethodThe ABCs of Russia's post-transition growth: What went wrong?Preprint
Ogloblin, C. G., & Brock, G. J. (2024) · Research Square · doi:10.21203/rs.3.rs-3871078/v1
Sari, S. P., Romdhon, M. M., & Utama, S. P. (2024) · Agroindustrial Journal · doi:10.22146/aij.v10i2.90178
The Impact of News Coverage in the Host Country on the Efficiency of China's Outward FDIPreprint
Yang, L., Du, W., & Luo, S. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4742714
Ibrahim, R., Hamidu, B. A., Kikelomo, B. O., et al. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4770193
Hailu, G., Weersink, A. J., & Minten, B. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4774330
Cai, J., Horrace, W. C., & Lee, Y. (2024) · Econometric Reviews · doi:10.1080/07474938.2024.2328905
Sustainable intensification of smallholder farming systems in Sub-Saharan Africa and South Asia
Karikari, B., Tetteh, F., Kizito, F., Folorunso, M. A., Mutungi, C., Kotu, B. H., Ansah, T., & Rahman, N. A. (2024) · Frontiers research topics · doi:10.3389/978-2-8325-4647-5
Innovation and Productivity Dynamics in the Service Sector with EndogeneityPreprint
Kumbhakar, S. C., Alemayehu, F. K., & Sun, K. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4814316
Bansal, R., & Maiti, D. (2024) · Research Square · doi:10.21203/rs.3.rs-4338896/v1
Wicaksono, D. S., & Budiasih, B. (2024) · Research Square · doi:10.21203/rs.3.rs-4451169/v1
RetractedMebratu, T., & Muleta, G. (2024) · Journal of Innovative Research · doi:10.54536/jir.v2i2.2702
“In assessing the level of efficiency, the two-step estimation procedure is reported to be biased (Battese & Coelli, 1995; Wang & Schmidt, 2002), and the single estimation method was followed to asses both the profit and the stochastic frontier profit inefficiency simultaneously using the maximum…”
MethodDoes workforce affect technical efficiency? A Stochastic Frontier Approach across industriesPreprint
Pinheiro, C. M., & da Silva, P. P. (2024) · Research Square · doi:10.21203/rs.3.rs-4469013/v1
Sources of Technical Inefficiency for Smallholder Field Pea Producers in Ethiopia
Muleta, G. (2024) · American Journal of Multidisciplinary Research and Innovation · doi:10.54536/ajmri.v3i4.2761
“This will overcome the estimation bias of the widely used two-step approach (Battese and Coelli, 1995; S. Kumbhakar et al. , 1991; S. C. Kumbhakar et al. , 2015; Wang and Schmidt, 2002).”
MethodDynamic evolution of utilization efficiency of medical and health services in China
Zhang, J. (2024) · PLoS ONE · doi:10.1371/journal.pone.0304157
Trang, N. T., Đức, D. T., & Tú, V. H. (2024) · The Journal of Environment & Development · doi:10.1177/10704965241266690 · Cited by 1
Capital Inflow and Technical Efficiency: A Comparative Analysis between India and China
Bansal, R. (2024) · Journal of Asian Economic Integration · doi:10.1177/26316846241266709
Determinants of Labour Demand in Manufacturing Sector in Ethiopia
Wajebo, T. W. (2024) · Indian Journal of Labour Economics · doi:10.1007/s41027-024-00515-3
“An augmented Cobb–Douglass production function is adopted for this study because it gives room to consider all-important explanatory variables identified in the empirical literature (Wang & Schmidt 2002; De Mello 1999).”
Rauf, R. I., Ayinde, K., Hamidu, B. A., et al. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4935832
ICTs and Industrial Efficiency under Borrowing Constraints in South AfricaPreprint
Mazorodze, B. T. (2024) · Research Square · doi:10.21203/rs.3.rs-5144436/v1
A novel ranking method in data envelopment analysis: a real case on Chinese banking industry
Nematizadeh, M., Amirteimoori, A. A., Kordrostami, S., et al. (2024) · Journal of Modelling in Management · doi:10.1108/jm2-04-2024-0122
Kumbhakar, S. C., Sickles, R. C., & Wang, H. (2024) · Advanced studies in theoretical and applied econometrics · doi:10.1007/978-3-031-48385-1_1
Self-citationReferences and Further Reading
(2025) · Emerald Group Publishing Limited eBooks · doi:10.1108/978-1-83797-743-720251011
Incorporating Heterogeneity in the 2TSF Model
Papadopoulos, A., & Parmeter, C. F. (2025) · Springer eBooks · doi:10.1007/978-3-031-81513-3_8
Does pollution prevention enhance firm efficiency? Insights from U.S. manufacturing firms
Memon, S. F., Kumbhakar, S. C., & Su, Y. (2026) · International Journal of Industrial Organization · doi:10.1016/j.ijindorg.2026.103305
Lambarraa-Lehnhardt, F., Wals, M., Raggas, A., et al. (2025) · Journal of Agriculture and Food Research · doi:10.1016/j.jafr.2025.101986
Metadata
- Authors
- Wang, H.-J., & Schmidt, P.
- Affiliations
- Institute of Economics, Academia Sinica (TW)
- Michigan State University (US)
- Venue
- Journal of Productivity Analysis
- Published
- 2002-09
- Volume
- 18
- Issue
- 2
- Pages
- 129-144
- ISSN
- 0895-562X (ISSN-L)
- 1573-0441
- OpenAlex
- W1524136776
- Semantic Scholar
- 0ba29f8abcbf8683e8286f2af42fa42ac20b1c6f
- Cited by
- 1,045
- References
- 10
- Citations by year
- 2026: 42 · 2025: 56 · 2024: 63 · 2023: 65 · 2022: 61 · 2021: 74 · 2020: 63 · 2019: 64 · 2018: 70 · 2017: 51 · 2016: 48 · 2015: 55 · 2014: 49 · 2013: 39 · 2012: 34
- FWCI
- 17.56
- Type
- journal article
- Language
- en
- Open access
- Licence
- Locations
- Journal of Productivity Analysis (publishedVersion)
- Munich Personal RePEc Archive (Ludwig Maximilian University of Munich) (submittedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- Topics
- Social Sciences › Decision Sciences › Management Science and Operations Research › Efficiency Analysis Using DEA
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Economic Growth and Productivity
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Spatial and Panel Data Analysis
- Keywords
- stochastic frontier analysis, technical efficiency, firm characteristics, one-step estimation, maximum likelihood estimation, Monte Carlo simulation
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